Question

求证：cosx+cos2x+…+cosnx=cosn+12x?sinn2xsinx2

求证：cosx+cos2x+…+cosnx=cosn+12x?sinn2xsinx2．

Answer 1

解答：证明：∵2sin

x

2

cosnx=sin(

x

2

+nx)+sin(

x

2

?nx)．
∴2sin

x

2

（cosx+cos2x+…+cosnx）=（sin

3x

2

?sin

x

2

）+(sin

5x

2

?

3x

2

)+…+(sin

1+2n

2

x?sin

1?2n

2

x)
=sin

1+2n

2

x?sin

x

2

=2cos

n+1

2

xsin

n

2

x．
∴cos+cos2x+…+cosnx=

cos n+1 2 x?sin n 2 x

sin x 2

．

Answer 2